A recent study conducted by Penn, Shone, and Borland, on...

A recent study conducted by Penn, Shone, and Borland, on behalf of LastMinute .com, revealed that $52 \%$ of business travelers plan their trips less than two weeks before departure. The study is to be replicated in the tri-state area with a sample of 12 frequent business travelers. a. Develop a probability distribution for the number of travelers who plan their trips within two weeks of departure. b. Find the mean and the standard deviation of this distribution. c. What is the probability exactly 5 of the 12 selected business travelers plan their trips within two weeks of departure? d. What is the probability 5 or fewer of the 12 selected business travelers plan their trips within two weeks of departure?

A recent study conducted by Penn, Shone, and Borland, on behalf of LastMinute .com, revealed that $52 \%$ of business travelers plan their trips less than two weeks before departure. The study is to be replicated in the tri-state area with a sample of 12 frequent business travelers.
a. Develop a probability distribution for the number of travelers who plan their trips within two weeks of departure.
b. Find the mean and the standard deviation of this distribution.
c. What is the probability exactly 5 of the 12 selected business travelers plan their trips within two weeks of departure?
d. What is the probability 5 or fewer of the 12 selected business travelers plan their trips within two weeks of departure?

Key Concepts

Binomial Distribution

This concept pertains to a probability model that describes the number of successes in a fixed number of independent and identical trials, where each trial has only two possible outcomes (often termed as success and failure). It is applicable when each trial has the same probability of success, making it a fundamental model in discrete probability theory and statistics.

Probability Mass Function

The probability mass function (PMF) for a binomial distribution gives the probability of obtaining exactly a specified number of successes in the fixed number of independent trials. It is typically expressed as P(X = k) = C(n, k) p^k (1-p)^(n-k), where C(n, k) is the binomial coefficient. This formula is essential in determining the likelihood of particular outcomes in a discrete binomial experiment.

Mean of Binomial Distribution

The mean of a binomial distribution is calculated as the product of the number of trials (n) and the probability of success (p) for each trial. This parameter represents the expected number of successes and provides a measure of the center or average outcome in a binomial setting, which is crucial in understanding the distribution's behavior.

Standard Deviation of Binomial Distribution

The standard deviation in a binomial distribution is derived from the formula sqrt(n*p*(1-p)). It measures the spread or variability of the distribution around its mean. Understanding the standard deviation helps quantify the typical deviation of outcomes from the expected value, which is important in both statistical inference and decision-making.

Cumulative Probability

Cumulative probability in the context of a discrete distribution, such as the binomial distribution, is the probability that the random variable takes on a value less than or equal to a specified number. It is obtained by summing the individual probabilities from the PMF for all relevant outcomes. This concept is often used to assess probabilities over a range of outcomes rather than a single point estimate.

Transcript

00:01 All right.

00:01 So we're looking at a study conducted by this group on behalf of this website, lastman .com.

00:08 They found that 52 % of business travelers plan their chips less than two weeks before they go.

00:14 So they wanted to replicate or a study was conducted to replicate in the tri -state area.

00:22 Sorry.

00:23 A study is to be replicated in the tri -state area, sampling 12 business travelers.

00:28 And we need to develop a probability distribution for the number of travelers who plan their trips within two weeks of departure for these 12 people.

00:40 And find the mean, standard deviation of said distribution and the probability of the probabilities of exactly five or and the probability of five or fewer of those 12.

00:54 So let's get going there, people.

00:57 So the distribution, sentence of binomial distribution, because it's a clearly defined probability for each of the travelers, 52%, and it could be, it's binomial, meaning you could have one or two outcomes, and each trial is independent of one another.

01:18 So we're good to go with binomial assumptions.

01:23 We're using the binomial probability distribution.

01:26 So we're going to use our formula.

01:29 I have it written down here in a very clunky code or clunky notation, but it does what we're going to do.

01:41 That combination times the probability to the x power times the complement of that probability to the n minus x where n is the number.

01:52 So here we go.

01:53 So the combination of n, which is 12, and it tells us we know end is 12 because, comma x so x is going to change from zero to 12 so on that we do times the probability 0 .52 to the x power times one minus the probability 0 .52 all raised to the n minus x power so 12 minus that cell's power and there we go and it's nice to see this i encourage you to try this sum those probabilities and they should equal one because that's what a probability distribution is.

02:38 It's all probabilities and all probabilities in any given event, some to one.

02:43 So there we go.

02:46 Now the mean is given as some of that.

02:51 So the mean, mu is equal to n times pi, pi being the probability, not the ratio of the circle circumference to its diameter.

03:04 So n is 12 times.

03:07 0 .52 and there we go.

03:11 6 .24.

03:13 So closer to 6, you know, between 6 and 7, but leaning on the 6 side.

03:19 The standard deviation is the square root of the variance...